Ask Your Instructor
The problem with tests, especially academic ones, is that only the test developer knows why they think a given answer is correct. In a school context, the only way to know for sure is to ask your instructor.
All the terms you can choose relate to how activities are measured, rather than defining how the two tasks interrelate from a dependency/successor standpoint. That may be part of the problem with the test question.
At first glance, finish-to-start seems right to me too. My assumption is that the completion of planning is an essential prerequisite to starting the actual testing. That also seems to align with what the QA manager is suggesting.
With that said, a possible reason for selecting finish-to-finish is if you are measuring for “as late as possible” in a project, where one work package needs to be completed before another package can finish, but where the start of each package is neither tightly-coupled nor intrinsic to the relationship. In such cases, a prerequisite package must be finished before its dependent package can finish too. That doesn’t sound like the use case you described, but there may be additional context in the exam or class lectures that wasn’t included in your question.
As one example of how context matters, if test planning is allocated exactly one week, and the testing itself has a fixed cycle time or due date, then you might want to work backwards to define the schedule. Scheduling the finish of a fixed quantity of test planning as late as possible based on the required finish date of a fixed quantity of testing might then make sense from a scheduling perspective, even if it seems counterintuitive from a dependency-mapping viewpoint.
Scheduling and dependency mapping are related, but aren’t interchangeable. While finish-to-start makes a lot of sense for immediate follow-on activities, finish-to-finish measurements are often useful in scheduling deliberate delays or postponing dependencies. That may be what your instructor has in mind here.
Addressing Exam Errors
Of course, the question text could contain errors or omissions. The answer key could also simply be wrong, either through error or faulty test design.
In an academic setting, ask your instructor to clarify the reason for the selected answer. For a normed exam, poor quality or beta questions will hopefully be factored out if the scoring, or at least have minimal impact, even if the exam format doesn’t allow you to dispute incorrect or flawed items.